#7919674

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4523 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(2,6)(7,8)(9,10)(11,13)(12,20)(14,16)(15,18)(17,22)(19,21)(23,24)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,19)(13,16)(14,18)(17,21)(20,23)(22,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(10,16)(11,18)(17,20)(19,24)(25,26,30,29)(27,33,35,31)(28,32,37,36)(34,39,40,38);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,19)(13,16)(14,18)(17,21)(20,23)(22,24)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(1,3,5,4)(2,7,6,8)(9,12,14,22)(10,17,16,20)(11,19,18,24)(13,23,15,21)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,39,40,38);(1,4,5,3)(2,8,6,7)(9,13,14,15)(10,11,16,18)(12,21,22,23)(17,24,20,19)(25,27)(26,33)(28,34)(29,31)(30,35)(32,39)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=gh,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7826870);